deletions | additions       diff --git a/section_Filtering_and_Smoothing_label__.tex b/section_Filtering_and_Smoothing_label__.tex  index 265a354..2e019c5 100644  --- a/section_Filtering_and_Smoothing_label__.tex  +++ b/section_Filtering_and_Smoothing_label__.tex 
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results in values for the variances in $x$ and $y$, the correlation  $\rho$ between $x$ and $y$. The mean values $\mu$ were set to zero  (no-systematic bias). Figure~\ref{fig:measurementmodel} shows the  results of one such evaluation.\begin{figure}[h!]  \begin{center}  \includegraphics[width=0.7\columnwidth]{figures/measurement_model/default_figure}  \caption{{\label{fig:measurementmodel} Measurement model showing the delta x and delta y  positions%  }}  \end{center}  \end{figure}  This allows to make use of the information in all $k$ neighbors and  keep track of a multimodal distribution. While keeping track of a  multimodal distribution allows for incorporating several possible  states of the system, the problem arises of which mode is the best  one. Using a weighted average of the modes would again introduce the  problem, that the weighted average falls into a low density  region. Therefore, the maximum a posteriori estimate, as described in  \cite{driessen2008map} is used. This approach uses the following  formula to obtain the MAP estimate:  Therefore, the final position estimate is equal to the position of one  of the particles.  \begin{align}  s_k^{MAP} &= \argmax_{s_k}{p(s_k \mid Z_k)}\\  &= \argmax_{s_k}{p(z_k \mid s_k) p(s_k \mid Z_{k-1})}   \end{align}  Therefore, the MAP estimator in our case is  \begin{align}  s_k^{MAP} = \argmax_{s_k} \lambda^M(s_k)  \end{align}  with  \begin{align}  \lambda^M(s_k) = p(z_k \mid s_k) \sum_{j=1}^Mp(s_k \mid s_{k-1}^j)w^j_{k-1}  \end{align}  This function is now only evaluated at a finite, chosen number of  states, the particles, using  \begin{align}  \hat{s}_k^{MAP} = \argmax_{s_k \in \{s_k^i \mid i=1,\ldots,N\}} \lambda^M(s_k)  \end{align}  In this formula, $p(z_k \mid s_k)$ is the likelihood of the particle  $s_k$ given the current measurement $z_k$. In our setting, this  probability is equal to the weight of the particle $z_k$, therefore  $p(z_k \mid s_k) = w^i_k$.  The estimation of \emph{uncertainty} is a core part of the proposed  approach, being important for safety and accuracy. Therefore,  uncertainty was modeled using the spread of the particles.  In every time step, the particles of the filter get updated based on  the optical flow estimates. These estimates are noisy, as illustrated  in Figure~\ref{fig:edgeflow}. Additionally, optical flow estimates  aggregate noise over time, since each estimate is dependent on the  previous one, leading to drift (\emph{relative position  estimates}). In contrast, the machine learning-based method makes  independent predictions. While this allow for avoiding accumulating  errors, the predictions do not dependent on each other, and might be  `jumping' between two points. To combine the advantages of both  methods, and leverage out the disadvantages, the particle filter is  used.  An idea was to include the similarity to the neighbors as confidence  value, thus reducing the measurement noise, if a high similarity  between current histogram and a training histogram is  achieved. However, we found no correlation between these  variables. Figure~\ref{fig:cor_sim_measurement} displays the  dependence structure.